Hi I am a massive fanfic reader and I use http://storymaster.the-code-monkey.com it's so easy. Choose which fic site you want inc Twilighted put in story or author url, your email address and choose which mode you want the story to download in e.g Mobi for Kindle.

Storymaster then emails you the story I then use Calibre to transfer it to my Kindle.

Why have a story emailed to you when the ones listed here will just download straight to your computer?

That saves a step right there by not having to enter your email address and someone else collect it for you.

And there's one site http://archiveofourown.org/ that has fanfic you can download straight from the story page. You click "Download" and get the choice of the story in Mobi, Epub, PDF, and HTML.